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Lyapunov spectral intervals: Theory and computation. (English) Zbl 1021.65067
In this survey paper two well known researchers in the numerical computation of Lyapunov exponents have made a rather complete discussion of theoretical properties and approximate calculation of these numbers. The paper starts with a review of some results on Lyapunov exponents and the equivalence between stability of distinct Lyapunov exponents and integral separation. Then, the authors made clear that although Lyapunov exponents is a set of finite points, for linear non autonomous problems is more convenient to think such a set as a continuum. Three definitions of this spectra are considered: the original Lyapunov definition, the Sacker-Sell spectrum definition based on exponential dichotomy and a third one based upon the information obtained when using the \(QR\)–method to approximate Lyapunov exponents. In this context the authors investigate some relationships between the three definitions. Next, the two main techniques to approximate Lyapunov exponents namely the continuous \(QR\)–method and the smooth singular value decomposition are revisited and a discussion of how to verify the integral separation is presented. Finally the results of several numerical experiments are given.

MSC:
65P40 Numerical nonlinear stabilities in dynamical systems
37C75 Stability theory for smooth dynamical systems
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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